nature of particle-in-the-box solutions
TRANSCRIPT
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2.3 Particle in a box
Slides: Video 2.3.6 Nature of the
particle-in-a-box solutionsText reference: Quantum Mechanics
for Scientists and EngineersSection 2.6 (second part)
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The particle in a box
Nature of particle-in-a-box sol
Quantum mechanics for scientists and engineers Dav
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Eigenvalues and eigenfunctions
Solutionswith a specific set of allowed values
of a parameter (here energy)eigenvalues
and with a particular functionassociated with each such value
eigenfunctionscan be called eigensolutions
2
n
z
z L
2
2n E
m
1, 2,n
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Eigenvalues and eigenfunctions
Heresince the parameter is an energy
we can call the eigenvalueseigenenergies
and we can refer to theeigenfunctions as the
energy eigenfunctions
2
n
z
z L
2
2n E
m
1, 2,n
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Degeneracy
Note in some problemsit can be possible to have more than oneeigenfunction with a given eigenvalue
a phenomenon known as
“degeneracy”The number of such states with the same
eigenvalue is called“the degeneracy”of that state
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Parity of wavefunctions
Note these eigenfunctions havedefinite symmetrythe function is the mirror
image on the left of what it is onthe rightsuch a function has “even parity”
or is said to be an “even function”The eigenfunction is also even 1n
2n
3n 1n
3n
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Parity of wavefunctions
The eigenfunction is aninverted imagethe value at any point on the right
of the centeris exactly minus the value at the“mirror image” point on the left
of the centerSuch a functionhas “odd parity”or is said to be an “odd function”
1n
2n
3n
2n
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Parity of wavefunctions
For this symmetric well problemthe functions alternate between
being even and oddand all the solutions are eithereven or oddi.e., all the solutions have a
“definite parity”Such definite parity is common insymmetric problemsit is mathematically very helpful
1n
2n
3n
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Quantum confinement
This particle-in-a-box behavior isvery different from the classical case1 – there is only a discrete set of
possible values for the energy
2 – there is a minimum possibleenergy for the particle
corresponding toheresometimes called a
“zero-point energy”
1n
2n
3n
E n e r g y
01n 22
1 / 2 / z E m L
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Quantum confinement
3 - the particle is not uniformlydistributed over the box, andits distribution is different fordifferent energies
It is almost never found verynear to the walls of the box
the probability obeys astanding wave pattern 1n
2n
3n
E n e r g y
0
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Quantum confinement
In the lowest state ( ),it is most likely to be found
near the center of the boxIn higher states,
there are points inside thebox where the particle will
never be found 1n
2n
3n
E n e r g y
0
1n
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Quantum confinement
Note thateach successively higher energy
statehas one more “zero” in theeigenfunction
This is very common behavior in
quantum mechanics 1n
2n
3n
E n e r g y
0
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Energies in quantum mechanics
In quantum mechanical calculationswe can always use Joules as units of energy
but these are rather largeA very convenient energy unit
which also has a simple physical significanceis the electron-volt (eV)
the energy change of an electron in movinthrough an electrostatic potential changeEnergy in eV = energy in Joules/e
e – electronic charge (C
191.602 10 J
191.602176 565 10 C
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Orders of magnitude
E.g., confine an electron in a 5 Å (0.5 nm) thbox
The first allowed level for the electron is
The separation between the first andsecond allowed energies ( )is
which is a characteristic size of majoenergy separations between levels i
an atom
22 10 19
1 / 2 / 5 10 2.4 10 1.5Jo E m
2 1 13 E E E
4.5eV
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